Can't we use a little basic topology to proove this? We discussed homotopic curves or surfaces and I believe when for a force field F. [tex]\del X F = 0[/tex] then there exists one. I might be able to look this up, in our class actually we were showed the techniques of evaluating surface integrals, but not the rigors of the proofs.
I think you need to show that the integral of the vector field over the surface is equivalent to the divergence of the vector field over the volume bounded by the surface. Then, clearly, if the vector field is the curl of a vector then the volume integral is zero.